Dihedral group

Dihedral group
Definition : For n?3 the Dihedral group is :
D_n={s^j r^i } such that j ? {0,1} and i ? {0,1,2,?,n-1}
Geometrically it s the group of symmetry of regular n-polygon consisting a clock wise rotations about the origin with angle (r^i ) i ? {0,1,2,?,n-1} and reflections about the principal axes (a cross lines through the origin (sr^i ) )

Example ( 1 ) D_3={1,r,r^2,s,sr,sr^2 }
1 is an identity element :(1)
r is a clock wise rotations about the center with angle :(1? 2? 3)
r^2 is a clock wise rotations about the center with angle :(1? 3?2)
s is a reflection about the line L_1 : ( 2?3)
sr is a reflection about the line L_2 :(1?3)
sr^2 is a reflection about the line L_3 :(1?2)





Note : By the same way we can study the molecular vibration of CH_3

Example ( 2 ) : Describe the molecular vibration of Methane CH_4
We may be study D_4={1,r,r^2,r^3 ,s,sr,sr^2,sr^3 }

1 is an identity element which make all H constant :(1)
r is a clock wise rotations about the center with angle :(1?2? 3?4)
r^2 is a clock wise rotations about the center with angle ??(1?? 3) (2??4)
r^3 is a clock wise rotations about the center with angle :(1?4?3?2)
s is a reflection about the line L_1 ?(2?4 )
sr is a reflection about the line L_2 ?(1?2)(3?4)
sr^2 is a reflection about the line L_3 ?(1?3 )
sr^3 is a reflection about the line L_4 ?(1?4)(2?3)
Example ( 3 ) : Discuss the molecular vibration of Arsenic pentachloride AsCl_5
We may be study D_5={1,r,r^2,r^3,r^4 ,s,sr,sr^2,sr^3,sr^4 }

1 is an identity element which make all Cl constant
r is a clock wise rotations about the center with angle :(1?2? 3?4?5)
r^2 is a clock wise rotations about the center with angle :(1?3? 5?2?4)
r^3 is a clock wise rotations about the center with angle :(1?4? 2?5?3)
r^4 is a clock wise rotations about the center with angle :(1?5?4?3?2)
s is a reflection about line L_1 ?(2?5)(3?4)
sr is a reflection about line L_2 ?(1?2)(3?5)
sr^2 is a reflection about line L_3 ?(1?3)(4?5)
sr^3 is a reflection about line L_4 ?(1?4)(2?3)
sr^4 is a reflection about line L_5 ?(1?5)(2?4)
Question: Discuss the molecular vibration of Benzene H_6 C_6.